[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-143486-105":59,"doc-detail-143486-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","mental-arithmetic-tricks-learn-number-tricks-for-fast-multiplication-and-roots","Mental Arithmetic Tricks - Learn number tricks for fast multiplication and roots","","A structured set of mental-arithmetic techniques for fast calculations, including multiplication tricks for specific numbers (such as 37 and 25), mirror-based products, and quick use of algebraic identities. It then extends to power and root tricks: learning cubes and cube roots from selected digits, determining square and fifth roots via modular patterns and leading-digit tables, and handling large-number cube roots with memorized remainder tables. The material emphasizes short explanations suitable for classroom use and demonstrates worked examples to verify results.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/mental-arithmetic-tricks-learn-number-tricks-for-fast-multiplication-and-roots/143486/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/mental-arithmetic-tricks-learn-number-tricks-for-fast-multiplication-and-roots/143486.png","ImageObject",300,407,{"name":92,"@type":93},"Aria","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-22","2026-08-25",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",23,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What are the key multiplication tricks taught in the document?","Question",{"text":113,"@type":114},"The document presents specific fast methods for cases like 37×3 leading to 111, and 25×4 leading to 100, along with mirror-based multiplication strategies that rely on digit properties.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How does the audience determine a two-digit cube root after hearing n³?",{"text":118,"@type":114},"It uses the fact that the last digit of x³ modulo 10 is a bijection for {0..9}, so the last digit of n can be inferred from n³, while the first digit comes from the magnitude bounds of n³.",{"name":120,"@type":111,"acceptedAnswer":121},"How are cube roots of large numbers handled in the advanced section?",{"text":122,"@type":114},"The approach combines memorized remainder/power tables to infer first and last digits, then determines middle digits by computing the value modulo 11 or modulo 100 using digit alternating sums and solving a congruence.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},143486,1787695388,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":24,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":130,"read_time":144},2336464648322,"https://ap-avatar.wpscdn.com/avatar/2200025388227c56fec?_k=1778556882303663488","Mental Arithmetic Tricks  \nAndreas Klein  \nThe normal tuxmath questions a simple. With the options TRICK_ \u003Ctrickname> _ALLOWED you can enable some tricky problems which need explanation.  \nThe first five tricks are suitable for disciples at 5th or 6th grade. I add a short explanation for each trick. Teachers should work out longer explanations with more examples before presenting them to the class.  \n1. The trick with the 37 (TRICK_37_ALLOWED)  \n37 × 3 = 111 . This give us 37 × (3n) = nnn.  \nExample:  \n37 × 12 = 444  \n12 ÷ 3 = 4  \n2. The trick with the 25 (TRICK_37_ALLOWED)  \n25 × 4 = 100 . This makes it easy to multiply 25 with any other two digit number.  \nExample:  \n25 × 35 = 8 75  \n35 ÷ 4 = 8 remainder 3, 3 × 25 = 75  \n3. The mirror at 5 (TRICK_MIRROR_5_ALLOWED)  \nFor the next trick we need two numbers with the property that the first  \ndigits are the same and the sum of the last digits is 10 .  \nThen it easy to calculate the product  \n(ab)10 × (ac)10 = (xy zw)10 a × (a + 1) = (xy)10  \nb × c = (zw)10  \nExample:  \n62 × 68 = 42 16  \n6 × 7 = 42  \n2 × 8 = 16  \n(Important: if the last two digits are 1 and 9, you must write 1 × 9 = 09 don’t forget the leading 0!)  \nThe proof of the trick is just a little exercise in calculus.  \n4. The mirror at 10 (TRICK_MIRROR_ 10_ALLOWED)  \nThe thrid binomial theorem (x + y)(x − y) = x2 − y 2 give us a nice trick for mental arithmetic, if we specialise x to a multiple of 10 and y to a number less than 10 .  \nExample:  \n76 × 64 = 48 64  \n7 × 7 − 1 = 48  \n6 × 6 = 36, 100 − 36 = 64  \n5. Squares (TRICK_SQUARE20 _ALLOWED)  \nThis is not really a trick, but it is useful to know the squares up to 20 .  \nThe next tricks are designed to make a show out of your mental arithmetic skills.  \n1. Exercising third powers (TRICK_CUBES_ALLOWED) You must learn the third power for the numbers from 1 to 10 .  \n x  1  2  3  4  5  6  7  8  9  10  x3  1  8  27  64  125  216  343  512  729  1000  \n2. Cube roots (TRICK_CUBEROOT_ALLOWED)  \nAsk one in the audience to choose a two digit number n and compute n3 . When you hear n3 you can tell immediately n.  \nThe trick is quite easy x 7→ x3 mod 10 is a bijection on {0 , . . . , 9} (see table above) . You know all thrid powers, so you get directly the last digit. The first digit you get from the size of the number.  \nExample  \nYou hear 658503 . Remove the last three digits. You get 658, since 83 ≤  \n658 \u003C 93 the result must be between 80 and 89 .  \nSince last digit is 3 and 73 = 343 the last digit of the result must be 7 . So you have the answer 87 .  \n3. 5th roots (TRICK_FIFTH_ROOT_ALLOWED)  \nTaking the 5th root is even easier than the cuberoot. You have x5 ≡ x mod 10, so the last digit is simple.  \nFor the first digit you learn the following table: x  Size of x5  \n1  100 thousand  \n2 3 million  \n3 24 million  \n4 100 million  \n5 300 million  \n6 777 million  \n7 1.6 milliards  \n8 3 milliards  \n9 6 milliards  \n10 10 milliards (Our American friends must replace the word milliards by billions.) You hear 8587340257 . As soon as you hear 8 milliards you know by the table that the first digit must be 9 . You ignore the rest and wait for the last digit which is 7, so the result is 97 .  \n4. Square roots (TRICK_SQUAREROOT_ALLOWED)  \nYou can also take fast square roots, if you know that the solution must bean integer. Consider the problem √7569.  \nSince 82 ≤ 75 \u003C 92 you know that the fist digit must be an 8 .  \nx2 ≡ 9 mod 10 has two solutions x = 3 and x = 7, so the last digit is either 3 or 7 . To decide which one is correct you calculate 852 = 7225 (use the trick mirror at 5) . 7569 > 7225, so the last digit must be 7 .  \nSolution √7569 = 87 .  \n5. Cube roots of large numbers (TRICK_LARGE_CUBEROOT_ALLOWED) Now it gets really hard. Ask one of the audience to compute the cube of a three digit number. You will tell the cube root.  \nI explain two ways to solve the problem  \nFirst method:  \nLearn the following table.  \n x  0  1  2  3  4  5  6  7  8  9  10  x3 mod 11  0  1  8  5  9  4  7  2  6  ","cbCaimhExxs0J7UZ","https://ap.wps.com/l/cbCaimhExxs0J7UZ","pdf",51700,"English","# Mental Arithmetic Tricks\n## Multiplication tricks (37, 25)\n## Mirror tricks (at 5 and 10)\n## Squares up to 20\n## Third powers and cube roots\n## Fifth roots and square roots\n## Cube roots of large numbers","[{\"question\":\"What are the key multiplication tricks taught in the document?\",\"answer\":\"The document presents specific fast methods for cases like 37×3 leading to 111, and 25×4 leading to 100, along with mirror-based multiplication strategies that rely on digit properties.\"},{\"question\":\"How does the audience determine a two-digit cube root after hearing n³?\",\"answer\":\"It uses the fact that the last digit of x³ modulo 10 is a bijection for {0..9}, so the last digit of n can be inferred from n³, while the first digit comes from the magnitude bounds of n³.\"},{\"question\":\"How are cube roots of large numbers handled in the advanced section?\",\"answer\":\"The approach combines memorized remainder/power tables to infer first and last digits, then determines middle digits by computing the value modulo 11 or modulo 100 using digit alternating sums and solving a congruence.\"}]","Mental Arithmetic Tricks - Learn number tricks for fast multiplication and roots | PDF",13]